Abstract: Expansions or ``addition theorems'' for the spherical wave functions , , and , with reference to the origin , have been obtained in terms of spherical wave functions with reference to the origin , where , has the coordinates with respect to .

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A similar integral representation for has been obtained by Satô, Yasuo, Bull. Earthquake Res. Institute, Tokyo, 28, 1-22 and 175-217 (1950). In this paper Yasuo obtains an addition theorem for also; however, his expansion is valid only for a translation of the origin along the -axis. The addition theorem he obtains contains powers of instead of the Bessel functions , and the coefficients of his expansion are not given by a general formula but must be evaluated by a recurrence relation.

A similar integral representation for has been obtained by Satô, Yasuo, Bull. Earthquake Res. Institute, Tokyo, 28, 1-22 and 175-217 (1950). In this paper Yasuo obtains an addition theorem for also; however, his expansion is valid only for a translation of the origin along the -axis. The addition theorem he obtains contains powers of instead of the Bessel functions , and the coefficients of his expansion are not given by a general formula but must be evaluated by a recurrence relation.